Standardized method of quantum state verification based on optimal strategy

ABSTRACT

The invention discloses a standardized method of quantum state verification based on optimal strategy. The specific steps are as follows: (1) Adjust the quantum device to generate the quantum states required; (2) Calculate the measurement basis under the optimal verification strategy; (3) The quantum device generates the quantum state copy by copy, and the optimal measurement basis is performed for each copy. The measurement results are recorded as 1 for success and 0 for failure; (4) Make statistics on the index of first failure N first  and the number of success events m pass  in N measurements; (5) Estimate the confidence and fidelity of the target state generated by the equipment according to the statistical results, and evaluate and analyze the reliability of the equipment. The invention realizes the standardized verification of the reliability of the quantum device, and estimates the quantum state with fewer resources.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to Chinese Patent Applications No.CN202010475173.4, filed on May 29, 2020. The content of theaforementioned applications, including any intervening amendmentsthereto, are incorporated herein by reference.

TECHNICAL FIELD

This disclosure generally belongs to the field of quantum stateverification in quantum information, and specifically relates to thestandardization of an optimal verification strategy in the reliable testof actual quantum equipment.

BACKGROUND

Quantum equipment for generating quantum states is an important modulein quantum information technology, which is used to generatesingle-particle states and multi-particle entangled states and widelyused in quantum communication, quantum simulation, quantum computing andother fields. Now there are many mature quantum devices for generatingquantum states, which are applied to various fields of quantumcommunication and quantum computing.

Checking whether a quantum equipment can reliably and effectivelygenerate the quantum state required by the customer is an important steptowards the large-scale application of quantum devices. As an end user,after receiving the quantum equipment, his/her hope is to adjust itsparameters to generate the quantum state that he/she needs. But inactual application scenarios, the device structure is not 100% perfect,and there will be various noises in operation, which will cause thequantum state actually generated by the device to be different from thetarget state needed by the customer. The goal of the customer is to useas few resources as possible to determine with a certain confidence thatthe device has produced the target state within a certain fidelity.

The traditional standardized method for characterizing quantum statesgenerated by a quantum equipment is quantum state tomography. However,as the number of particles and qubits increases, the number ofmeasurement bases required for quantum state tomography increasesexponentially. In addition, it requires tens of thousands of copies ofquantum states. To reduce statistical errors, the maximum likelihoodestimation is also used to obtain the density matrix in the datapost-processing. Due to the large amount of measurement settings anddata required for processing, this method is very time-consuming andresource-consuming.

In recent years, some non-tomographic methods have been proposed toverify the quantum state. They do not need to know the exact densitymatrix of the quantum state. They can estimate the quantum state with aspecific confidence level and fidelity level. However, these methodseither make some specific assumptions about the quantum state orrestrict specific measurement operations, which are not easy toimplement in practical applications. So far, no unified standardizedmethod like quantum state tomography has been established.

To this end, we have established a standardized quantum stateverification procedure by which to verify the quantum device withoptimal strategy. This procedure is universal and can be used for theverification of quantum products for generating quantum states in thefuture.

SUMMARY

In order to overcome the shortcomings of traditional quantum statetomography, the invention provides a set of standardized procedures forquantum state verification based on optimal verification strategies,standardizes the schemes for quantum state verification, and fully andefficiently analyzes quantum equipment. The technical scheme is asfollows:

A standardized method of quantum state verification based on optimalstrategy, including the following specific steps:

(1) Generating target state: Regarding different physical ensembles suchas ions, superconductors, photons, NV color centers, etc., adjusting thevarious components of the quantum equipment to generate the target staterequired by the customer.

Firstly optimize the phase of each module, monitoring the contrast ofeach channel in the standard base through the coincidence countsdetected by the single photon detector, and adjust the phase to maximizethe contrast in the standard base. The target state can be generated byadjusting the relative intensity and phase of different components inthe quantum state.

According to the form of the target state |ψ(r,ϕ)

required by the customer, adjust the different components in the quantumequipment so that the corresponding intensity and relative phase of thegenerated state are close to the target intensity and phase set by thecustomer, such that the device can work in the target state.

(2) Obtain the projective measurement required by the optimal strategy:Calculate the density matrix corresponding to the target state |ψ(r,ϕ)

by programming, so as to obtain the estimated values of the parameters rand ϕ in the target state.

For a general entangled state, the theory gives the projectivemeasurement required by the optimal strategy. The measurement basis isrelated to the values of r and ϕ in the target state. The values of theparameters r and ϕ estimated above can be used to calculate themeasurement basis corresponding to the projective measurement. Themeasurement basis is realized by a quantum state analyzer, which canperform both the non-adaptive and adaptive measurements. The adaptivemeasurement is realized using triggering instrument.

In practice, selecting multiple sets of parameters r and ϕ for thetarget state, then writing an automated calculation program. For eachgiven target state, the settings corresponding to the measurement basesin the quantum state analyzer can be obtained through the parameters rand ϕ.

(3) Realization of projective measurement: The quantum state is measuredby the state analyzer. This method uses both non-adaptive measurementand adaptive measurement, and the two measurements cooperate to realizea comprehensive evaluation for the quantum equipment.

Taking the two particle systems A and B as an example. Non-adaptivemeasurement does not require communication between A and B, and eachperforms local projective measurement. Adaptive measurement requiresclassical communication between A and B. The result of one party istransmitted to the other party in real time, and the triggeringinstrument of the other party is controlled to switch to thecorresponding measurement base, so as to realize the adaptivemeasurement with the help of classical communication.

The analyzer finally uses the time-correlated counter to detect theparticle, and records the coincidence count of each channel during themeasurement. The timestamp of each detection channel is obtained throughthe timetag technology in the optimal projective measurement basis.Writing a data processing program to separate and extract thecoincidence counts during a specific time window from the timetag file.Under each projective measurement basis, the strategy will have asuccess probability corresponding to specific coincidence counts. If theprojection occurs at two successful coincidence channels within thecoincidence window, the measurement result is recorded as success 1,otherwise recorded as failure 0.

(4) Statistics on the measurement results: Based on the optimalverification strategy, this invention method uses two cooperativemechanisms to ensure the reliability of verification.

Task A: Select the projective base from the measurement setssequentially. Each measurement is randomly selected according to theprobability of the projective base. The final measurement resultsconstitute a binary string 1111110 . . . , and record the position ofthe first failure event 0 as N_(first), and each N_(first) has aprobability of occurrence Pr (N_(first)), the cumulative probability ofsuccess for the previous n_(exp) measurements is:

$\delta_{A} = {\sum\limits_{N_{first} = 1}^{n_{\exp}}{\Pr\left( N_{first} \right)}}$

This gives the confidence of the target state generated by the device,and a desired confidence level δ_(A) can be taken to obtain the requirednumber of measurements n_(exp), which is the number of copies of quantumstates consumed to reach the δ_(A) confidence level.

Task B: Do a fixed number of measurements N, the statistical results ofthe measurements form a binary string 110101110 . . . 1, from which thenumber of success events m_(pass) is obtained. In theory, there will bea success probability μ≡1−Δ_(∈) related to the infidelity E of thetarget state. According to the relative magnitude of m_(pass) and μ, theequipment is classified into two cases, Case 1 (m_(pass)>μN) and Case 2(m_(pass)<μN), which belong to the inner region and the outer region ofa circle with radius ∈, respectively. The confidence of the equipmentcan be upper bounded using the Chernoff bound:

$\delta \equiv e^{- {{ND}({\frac{m_{pass}}{N}{}\mu})}}$

where

${D\left( {x{}y} \right)}:={{x\;{\log_{2}\left( \frac{x}{y} \right)}} + {\left( {1 - x} \right){\log_{2}\left( \frac{1 - x}{1 - y} \right)}}}$

is the Kullback-Leiber divergence. Finally, the confidence of δ_(B)=1−δcan be used to determine whether the device belongs to Case 1 or Case 2.

(5) Estimation and analysis of the confidence and fidelity: For task A,the copy index of quantum state where the first failure occursconstitutes a geometric distribution. N_(first)=n_(exp) means that theprevious n_(exp)−1 measurements are successful, while the n_(exp)-thmeasurement fails. The calculated cumulative probability is theconfidence that the device generates the target state. Therefore, thenumber of measurements n_(exp) obtained is the number required togenerate the confidence δ_(A). At the same time, one can estimate theinfidelity ∈_(exp) ^(Non) and ∈_(exp) ^(Adp) of the state produced bythe device by fitting the geometric statistics of probabilitydistribution. These infidelities corresponds to the estimation of theinfidelity of the quantum state obtained by non-adaptive and adaptivemeasurement, respectively.

For task B, a reasonable value of E based on the above fitted ∈_(exp)^(Non) and ∈_(exp) ^(Adp) and a fixed value of δ are given. According tothe formula of Chernoff bound, programming and calculating the variationof δ and ∈ along with the increase of the number of copies of quantumstates, and finally obtain the number of copies required for theconfidence to be δ_(B) and the scaling law of E versus N.

The advantages of this invention are that:

1. Compared with the traditional quantum state tomography method, thepresent invention requires fewer measurement bases. For example, for aqubit system, non-adaptive measurement requires four measurement bases,and adaptive measurement only requires three measurement bases.Moreover, the number of copies of the quantum state consumed is small,and a reliable estimation of the quantum state can be made withrelatively fewer copies. With the same number of copies, the presentmethod can achieve better accuracy than traditional quantum statetomography.

2. Compared with the existing quantum state verification and estimationschemes, the present invention provides a standardized workflow, relaxesthe strong assumptions in the original theoretical scheme. Consideringthe imperfect operation of the actual equipment, a comprehensivediscussion of its possible working conditions was given, which has agood practical and applicable prospect, and it could be used as astandardized method for checking the quantum equipment.

3. The data post-processing is simple and easy, and requires only simpleprogramming (such as matlab, mathematica, python, etc.) to get thevariation trend of the confidence and fidelity with the number ofmeasurements. In terms of the estimation of physical parameters, thescaling of ∈ versus N (∈˜N^(r)) can approach the Heisenberg limit ofr=1.

BRIEF DESCRIPTION OF THE DRAWINGS

Various embodiments in accordance with the present disclosure will bedescribed with reference to the drawings, in which:

FIG. 1 illustrates the schematic diagram of the working principle ofquantum state verification for the invention.

FIG. 2 illustrates the verification workflow of the invention.

FIG. 3 illustrates the data acquisition and processing of task A in theinvention.

FIG. 4 illustrates the data acquisition and processing of task B in theinvention.

FIG. 5 illustrates the setup of a verification example in the invention.

FIG. 6 illustrates the adaptive measurement implementation of theinvention.

DETAILED DESCRIPTION

In order to make the objectives, technical schemes and advantages of thepresent invention clearer, the present invention will be furtherdescribed below in conjunction with the accompanying drawings andspecific implementation examples. It should be noted that the specificembodiments described here are only used to explain the presentinvention, but not to be limited to the present invention.

As shown in FIG. 1, taking the photon system as an example, a quantumstate verification principle based on the optimal strategy is given. Theend user expects the quantum device to output the target state |ψ

. In actual working, the device is not perfect, and some states σ₁

σ₂

. . .

σ_(i)

. . .

σ_(N) that deviate from the target state will be generated in N times ofmeasurement. These states are called the copies of the target state. Foreach copy σ_(i) of the target state, the projective measurement baseM_(i) is randomly selected from the measurement sets {M₁, M₂, M₃, . . .} with the corresponding selection probability p_(i). Then themeasurement M_(i) is performed and the measurement result is recorded as1 for success and 0 for failure. The workflow of the verification isshown in FIG. 2.

Present invention uses two cooperative tasks to verify the quantumdevice. As shown in FIG. 3, task A counts the index of copies N_(first)where the first failure occurs. It is based on the followingassumption—the fidelity of the state σ_(i) produced by the device andthe target state of the device is either 1 or there is an infidelity ∈which is greater than 0 and the fidelity satisfies

ψ|σ_(i)|ψ

≤1−∈ for all states σ₁. The goal of task A is to distinguish these twocases. Since the target state is always the eigenstate of the projectiveoperator, it satisfies the test M_(i)|ψ

=|ψ

. In the worst case, the fidelity of the state generated by the deviceis less than 1∈, and the maximum probability of a passing the test is:

$\begin{matrix}{{\max\limits_{{({\psi{\sigma_{i}}\psi}\rangle} \leq {1 - \epsilon}}{{Tr}\left( {\Omega\sigma_{i}} \right)}} = {{1 - {\left\lbrack {1 - {\lambda_{2}(\Omega)}} \right\rbrack\epsilon}}:={1 - \Delta_{\epsilon}}}} & (1)\end{matrix}$

Among them, the measurement operator ψ=Σ_(i)p_(i)M_(i) is called averification strategy, Δ_(∈):=[1−λ₂(Ω)]∈ is the probability of failing asingle test, and λ₂(Ω) is the second largest eigenvalue of the Ωmeasurement operator. After N rounds of tests, the maximum probabilityof σ_(i) passing all tests in the worst case is (1−Δ_(E))^(N). In orderto obtain the confidence 1−δ, the minimum number of measurementsrequired is:

$\begin{matrix}{N \geq {\ln{\delta/{\ln\left\lbrack {1 - \Delta_{\epsilon}} \right\rbrack}}} \approx {\frac{1}{\Delta_{\epsilon}}\ln\frac{1}{\delta}}} & (2)\end{matrix}$

In order to minimize the consumption of measurement resources, thesecond largest eigenvalue λ₂(Ω) needs to be minimized. By optimizing thesecond largest eigenvalue, the projective measurement corresponding tothe optimal strategy can be obtained, which is called non-adaptivemeasurement strategy [Phys. Rev. Lett. 120, 170502 (2018)]. For a qubitquantum state, non-adaptive measurement requires four measurement bases{P₀, P₁, P₂, P₃}.

In order to obtain the optimal verification strategy for any quantumstate, a lemma is introduced: for any qubit state |104

, if its optimal strategy is Ω, then a target state connected by a localunitary operation |φ

=(U⊗V)|ψ

has the optimal verification strategy Ω(φ)=(U⊗V)Ω(U⊗V)^(†).

If classical communication is added, the number of measurements can bereduced. This is the optimal adaptive measurement strategy [Phys. Rev. A100, 032315 (2019)]. Adaptive measurement requires real-timecommunication between particles A and B. Considering the one-waycommunication from particle A to particle B, only three measurementbases {T₀, T₁, T₂} are required. T₀ is still the usual Pauli matrixmeasurement, and the realization of T₁ and T₂ requires the selection ofthe measurement operations at B's site in real time based on themeasurement results of A.

Considering that the actual equipment is not perfect, the more practicaltask (task B) is to give a threshold for the fidelity of the statesproduced by the device with a certain confidence. As shown in FIG. 4,considering two realistic situations, there exists a quantity ∈ greaterthan 0 such that:

Case 1: The equipment works correctly, and for any i, the fidelitysatisfies

ψ|σ_(i)|ψ

>1−∈.

Case 2: The equipment works incorrectly, and for any i, the fidelity is

ψ|σ_(i)|

≤1−∈.

For Case 1, there is a greater probability that the test will succeedand the number of successes is greater than the theoretical expectation.For Case 2, there is greater probability that the test will fail and thenumber of successes is less than the theoretical expectation. Accordingto the distribution of the number of successes m_(pass), whether thedevice belongs to Case 1 or Case 2 can be given with a certainprobability.

Next, the specific procedures of verification are given based on theabove principles, as shown in FIG. 2:

1. Adjust Quantum Devices to Produce the Desirable Quantum States

The quantum device has some tunable components for generating therequired quantum state. As shown in FIG. 5, the quantum light sourcegenerates a two-photon polarization entangled state. The form of thetarget state is:

|ψ(θ,ϕ)

_(AB)=sin θ|HV

+cos θe ^(iϕ) |VH

  (3)

The quarter-wave plate and half-wave plate in the quantum light sourcecan be adjusted to change the parameters θ and ϕ in the target state.The quantum light source pump periodically-poled potassium titanylphosphate (PPKTP) crystals bidirectionally to generate the entangledphoton pairs.

Parameterize the intensity parameter θ=k·π/10 in the target state, takesome discrete points k=1,2,3,4 with equal intervals, adjust the waveplate so that coincidence counts of HV and VH conform to the weightratio r=(sin θ/cos θ)². The density matrix of the quantum state isestimated by taking the count data accumulated for 1 second, and theoptimized phase ϕ is obtained through this density matrix.

2. Measurement Bases of Optimal Verification Strategy

By minimizing the second largest eigenvalue λ₂(Ω) corresponding to thestrategy Ω, the optimal measurement basis corresponding to the targetstate |ψ(θ, ϕ)

_(AB) can be obtained. The non-adaptive measurement has four projectivemeasurements, one of which is the ZZ measurement (particles A and B aremeasured by Pauli σ_(Z) operator):

P ₀ =|H

H|└|V∴

V|+|V

V|⊗|H

H|  (4)

The other three measurement bases P_(i)=|ũ_(i)

ũ_(i)|⊗|{tilde over (v)}_(i)

{tilde over (v)}_(i)|, whose expressions are as follows:

$\begin{matrix}{{\left. {\overset{˜}{u}}_{1} \right\rangle = {{\frac{1}{\sqrt{1 + {\tan\theta}}}\left. H \right\rangle} + {\frac{e^{\frac{2\pi i}{3}}}{\sqrt{1 + {\cot\theta}}}\left. V \right\rangle}}},{\left. {\overset{˜}{v}}_{1} \right\rangle = {{{\frac{1}{\sqrt{1 + {\tan\theta}}}\left. V \right\rangle} + {\frac{e^{\pi{i/3}}e^{i\;\phi}}{\sqrt{1 + {\cot\theta}}}{\left. H \right\rangle.\left. {\overset{˜}{u}}_{2} \right\rangle}}} = {{\frac{1}{\sqrt{1 + {\tan\theta}}}\left. H \right\rangle} + {\frac{e^{\frac{4\pi i}{3}}}{\sqrt{1 + {\cot\theta}}}\left. V \right\rangle}}}},{\left. {\overset{˜}{v}}_{2} \right\rangle = {{{\frac{1}{\sqrt{1 + {\tan\theta}}}\left. V \right\rangle} + {\frac{e^{5\pi{i/3}}e^{i\;\phi}}{\sqrt{1 + {\cot\theta}}}{\left. H \right\rangle.\left. {\overset{˜}{u}}_{3} \right\rangle}}} = {{\frac{1}{\sqrt{1 + {\tan\theta}}}\left. H \right\rangle} + {\frac{1}{\sqrt{1 + {\cot\theta}}}\left. V \right\rangle}}}},{\left. {\overset{˜}{v}}_{3} \right\rangle = {{\frac{1}{\sqrt{1 + {\tan\theta}}}\left. V \right\rangle} + {\frac{e^{3\pi{i/3}}e^{i\;\phi}}{\sqrt{1 + {\cot\theta}}}{\left. H \right\rangle.}}}}} & (5)\end{matrix}$

The expressions of the adaptive measurement bases {T₀, T₁, T₂} is in thefollowing:

T ₀ =|H

H|⊗|V

V|+|V

V|⊗|H

H|

T ₁=|+

+|⊗|{tilde over (υ)}₊

{tilde over (υ)}₊|+|−

−|⊗|{tilde over (υ)}⁻

{tilde over (υ)}⁻|

T ₂ =|R

R|⊗{tilde over (ω)} ₊

{tilde over (ω)}₊|+|−

−|⊗|{tilde over (ω)}⁻

{tilde over (ω)}⁻|  (6)

where,

$\begin{matrix}{{{\left.  + \right\rangle = \frac{\left. V \right\rangle + \left. H \right\rangle}{\sqrt{2}}},{\left.  - \right\rangle = \frac{\left. V \right\rangle - \left. H \right\rangle}{\sqrt{2}}}}{{\left. R \right\rangle = \frac{\left. V \right\rangle + {i\left. H \right\rangle}}{\sqrt{2}}},{\left. L \right\rangle = \frac{\left. V \right\rangle - {i\left. H \right\rangle}}{\sqrt{2}}}}} & (7)\end{matrix}$|{tilde over (υ)}₊

=e ^(iϕ) cos θ|H

+sin θ|V

,|{tilde over (υ)} ⁻

=e ^(iϕ) cos θ|H

−sin θ|V

|{tilde over (ω)}₊

=e ^(iϕ) cos θ|H

−i sin θ|V

,|{tilde over (ω)} ⁻

=e ^(iϕ) cos θ|H

+i sin θ|V

  (8)

The expressions of the measurement bases are quantities related to theparameters (θ, ϕ) in the target state. Using the Jones matrix method,programming and calculating the setting parameters in the quantum stateanalyzer corresponding to the above-mentioned projective bases, andrealizing the projective measurement for the polarized state.

3. Implementation of Projective Measurement

The device sequentially generates a series of copies of quantum statesσ_(i). In FIG. 5, the dotted boxes at both ends of A and B are themeasurements performed by the A and B photons. When the wave plate andelectro-optic modulator components in the adaptive measurement areremoved at the B's site, the non-adaptive measurement is performed.Non-adaptive measurement does not require classical communication. Usingthe parameters θ and ϕ of the target state, the angles of thequarter-wave plate and half-wave plate required to realize theprojective measurement {P₀, P₁, P₂, P₃} can be calculated according tothe expressions of |ũ_(i)

and |{tilde over (v)}_(i)

.

For adaptive measurement, B uses two electro-optic modulators to receivethe measurement results of A in real time, so as to realize theprojective measurement of {tilde over (υ)}₊{tilde over (υ)}⁻ and {tildeover (ω)}₊/{tilde over (ω)}⁻ 0 according to the measurement result of A.If the measurement result at A's site is |+

or |R

, the former electro-optic modulator performs the corresponding rotationoperation, and the latter electro-optic modulator maintains the identitymatrix transformation. If the measurement result at A's site is |−

or |L

, the latter electro-optic modulator performs the corresponding rotationoperation, and the former electro-optic modulator does the identityoperation.

The implementation diagram of the adaptive measurement is shown in FIG.6. Specifically, the electro-optic modulator 1 will convert thepolarization states of {tilde over (υ)}₊ and {tilde over (ω)}₊ to the Hpolarization state, and finally exit from the transmission port of thepolarized beam splitter (PBS) and enter the single photon detector.Correspondingly, {tilde over (υ)}⁻ and {tilde over (ω)}⁻ will be rotatedby the electro-optic modulator 2 to the V polarization state and comeout from the reflection port of the PBS. The measurement result of |+

/R

at A's site is used to trigger the response of the electro-opticmodulator 1 through the electrical signal, while the measurement resultof |−

/L

is used to trigger the response of electro-optic modulator 2. Only oneelectro-optic modulator is active at a time, and the other electro-opticmodulator performs identity operation. The specific operations ofadaptive measurement are shown in the following table:

Projective Measurement Measurement measurement setup of A Measurement ofB results Probability of success T₀ H I (Electro- optic I (Electro-optic —$\frac{{CC}_{HV} + {CC}_{VH}}{{CC}_{HV} + {CC}_{HH} + {CC}_{VV} + {CC}_{VH}}$modulator modulator 1) 2) V I I — (Electro- (Electro- optic opticmodulator modulator 1) 2) T₁ + {tilde over (υ)}₊ (Electro- optic I(Electro- optic {tilde over (υ)}₊ → H$\frac{{CC}_{+ {\overset{\sim}{\upsilon}}_{+}} + {CC}_{- {\overset{\sim}{\upsilon}}_{-}}}{{CC}_{+ {\overset{\sim}{\upsilon}}_{+}} + {CC}_{+ {\overset{\sim}{\upsilon}}_{-}} + {CC}_{- {\overset{\sim}{\upsilon}}_{+}} + {CC}_{- {\overset{\sim}{\upsilon}}_{-}}}$modulator modulator 1) 2) − I {tilde over (υ)}⁻ {tilde over (υ)}⁻ → V(Electro- (Electro- optic optic modulator modulator 1) 1) T₂ R {tildeover (ω)}₊ (Electro- optic I (Electro- optic {tilde over (ω)}₊ → H$\frac{{CC}_{R{\overset{\sim}{\omega}}_{+}} + {CC}_{L{\overset{\sim}{\omega}}_{-}}}{{CC}_{+ {\overset{\sim}{\omega}}_{+}} + {CC}_{R{\overset{\sim}{\omega}}_{-}} + {CC}_{L{\overset{\sim}{\omega}}_{+}} + {CC}_{L{\overset{\sim}{\omega}}_{-}}}$modulator modulator 1) 2) L I {tilde over (ω)}⁻ {tilde over (ω)}⁻ → V(Electro- (Electro- optic optic modulator modulator 1) 2)

The failure probability for single non-adaptive measurement isΔ_(∈)=1−∈/(2+sin θ cos θ), which is greater than the failure probabilityof single adaptive measurement Δ_(∈)=1−∈/(2−sin² θ). Therefore, toachieve the same confidence level of 1−δ, the number of copies ofadaptive measurement is less. That is to say, adaptive measurementconsumes fewer number of copies of quantum states at the expense ofallowing classical communication.

4. Statistics of Measurement Results, Data Extraction and Processing

The timetag data of single photon detector are extracted using the fieldprogrammable logic gate array. The file of timetag data is stored as twocolumns. The first column is the label of each detection channel, andthe second column is the response time stamp of the correspondingdetection channel. The processing program takes each time slice as aunit. The initial time is t_(i)=1. The time increases by graduallyskipping to the next line and finally reaches the end time t_(f). Onceone coincidence count is found between the t_(i) and t_(f) rows, thecorresponding coincidence channel is recorded and is treated as one copyof the quantum state. Next time starts with t_(i)=n, the coincidencecounts are scanned from t_(f)=n+1 until the next coincidence count isfound between t_(i) and t_(f). This process is iterated until all singlecoincidence counts are separated, and finally the projective results ofeach copy of quantum state can be obtained.

The measurement base is selected randomly according to the randomnumbers and the statistics of measurement results are made. Successevent is recorded as 1, and failure event is recorded as 0. Fornon-adaptive measurement, the probabilities that the four measurementbases {P₀, P₁, P₂, P₃} are selected are μ₀=(2−sin 2θ)/(4+sin 2θ),μ₁=μ₂=μ₃=(1−μ₀)/3, respectively. For adaptive measurement, theprobabilities of {T₀, T₁, T₂} being selected are

$\left\{ {{\beta(\theta)},\frac{1 - {\beta(\theta)}}{2},\frac{1 - {\beta(\theta)}}{2}} \right\},$

respectively, where β(θ)=cos² θ/(1+cos² θ). Whether success or failureof the measurement result can be determined according to the channelwhere the coincidence count occurs. For example, for non-adaptivemeasurement, the success probability of the four measurement bases is:

$\begin{matrix}{{P_{0}:\frac{{CC}_{HV} + {CC}_{VH}}{{CC}_{HH} + {CC}_{HV} + {CC}_{VH} + {CC}_{VV}}}{P_{i}:\frac{{CC}_{{\overset{˜}{u}}_{i}{\overset{˜}{v}}_{i}^{\bot}} + {CC}_{{\overset{˜}{u}}_{i}^{\bot}{\overset{˜}{v}}_{i}} + {CC}_{{\overset{˜}{u}}_{i}^{\bot}{\overset{˜}{v}}_{i}^{\bot}}}{{CC}_{{\overset{˜}{u}}_{i}{\overset{˜}{v}}_{i}} + {CC}_{{\overset{˜}{u}}_{i}{\overset{˜}{v}}_{i}^{\bot}} + {CC}_{{\overset{˜}{u}}_{i}^{\bot}{\overset{˜}{v}}_{i}} + {CC}_{{\overset{˜}{u}}_{i}^{\bot}{\overset{˜}{v}}_{i}^{\bot}}}}} & (9)\end{matrix}$

Where i=1, 2, 3. For P₀ projective measurement, if the coincidence countoccurs in CC_(HV) or CC_(VH), σ_(i) passes the test and the result isrecorded as 1. Otherwise if the coincidence count falls on otherchannels, the test fails and the result is recorded as 0. For P_(i)projective measurement, if a single coincidence count falls on

${CC}_{{\overset{\sim}{u}}_{i}{\overset{\sim}{v}}_{i}^{\bot}},{{CC}_{{\overset{\sim}{u}}_{i}^{\bot}{\overset{\sim}{v}}_{i}}\mspace{14mu}{or}\mspace{14mu}{CC}_{{\overset{\sim}{u}}_{i}^{\bot}{\overset{\sim}{v}}_{i}^{\bot}}},$

the measurement is recorded as success 1, otherwise measurement isrecorded as failure 0. For adaptive measurement, the measurement resultscan also be obtained according to the probability of success under eachprojective measurement.

The number of copies of the quantum state are gradually increased byprogramming, and the binary sequence 11101001 . . . 1 is obtainedthrough the result of the coincidence counts. The task A is performedand the index of the copy of quantum state is recorded. The firstoccurrence of 0 is labelled as N_(first). To reduce statistical error,10,000 rounds of repetitions are performed. The probability for theoccurrence of N_(first) is obtained. The task B is then performed byfixing the total number of measurements N. Also, 1000 rounds ofrepetitions is averaged to reduce the statistical error. The number ofsuccess events is recorded to obtain the number m_(pass) in the Nmeasurements.

5. Evaluate the Confidence and Fidelity of the Target State Generated bythe Equipment

Through task A, the number of copies of the quantum state required toreach 90% confidence can be calculated. That is, using the probabilityPr(N_(first)) of the first failure which occurs at N_(first), thecumulative probability can be obtained:

δ_(A)=Σ_(N) _(first) ₌₁ ^(n) ^(exp) Pr(N _(first))  (10)

Setting δ_(A)=90%, the value of n_(exp) can be calculated.

At the same time, the infidelity of the quantum state, i.e. ∈_(exp)^(Non) (non-adaptive) and ∈_(exp) ^(Adp) (adaptive), can be estimated byfitting the probability distribution of N_(first). This estimatedparameter is used as the foundation for selecting the ϵ parameter intask B. The theoretical expected success probability in task B is

=1−Δ_(∈)≡μ. By using the Chernoff bound formula:

$\begin{matrix}{\delta \equiv e^{- {{ND}({\frac{m_{pass}}{N}{}\mu})}}} & (11)\end{matrix}$

The device is classified as Case 1 or Case 2 by taking suitable E. UnderCase 1, the expected number of success events m_(pass)≥Nμ. The aboveformula can be used to calculate the variation of confidence δ_(B)=1−δwith the number of copies of the quantum state. Under Case 2, theexpected number of success events m_(pass)≤Nμ, and the variation ofconfidence in this region can also be obtained in the same way. When theconfidence level 1−δ is given, the Chernoff bound can be used tocalculate the variation of ϵ versus the number of copies of quantumstate N to obtain a scaling law ∈˜N^(r) for the estimation of infidelityparameters.

For the estimation of the confidence parameter and the infidelityparameter, this invention can achieve a better confidence and a higherfidelity under the same number of copies of quantum state.

The invention discloses a quantum state verification standardizationmethod based on an optimal strategy. The basic principles, main workingprocedures and advantages of the present invention are shown anddescribed above. Those people skilled in the industry should understandthat the invention is not limited by the above-mentioned embodiments.The above-mentioned embodiments and the specifications describe only theprinciples of the invention. Without departing from the spirit and scopeof the invention, the present invention will have various changes andimprovements.

The verification method disclosed in the present invention is notlimited to the photonic system, nor is it limited to the number ofphotons. It is suitable for various quantum systems such as ions,superconductors, and semiconductors. It only needs to select differentstrategies corresponding to the system specified to achievecorresponding device verification based on different platforms. Allthese changes and improvements fall within the scope of the claimedinvention.

We claim:
 1. A standardized method of quantum state verification basedon optimal strategy, comprising: Step
 1. The target state |ψ

is generated by adjusting the quantum device. The coincidence count ismeasured through the time-correlated detector module, and the weight andphase of the quantum state are adjusted through the instrumentsadjustable parameters. During the adjustment, the ratios of coincidencecount for different channels are varied, so that the weight of thetarget state and the coincidence count ratio are consistent as well asthe phase is compensated. Finally the instrument settings that producesthe target state are determined; Step 2: Record the coincidence countsunder Pauli's complete measurement base, optimize the density matrix ofthe target state, and estimate the value of the weight and phaseparameters in the target state. In this step, you can also directly setthe weight and phase of the target state required by the customer;Determine the projective measurements {p₁M₁, p₂M₂, . . . , p_(i)M_(i), .. . , p_(N)M_(N)} of the optimal strategy in the quantum state analyzer,and calculate the setting parameters of non-adaptive measurement (M_(i)

P_(i)) or adaptive measurement (M_(i)

T_(i)) required in the quantum state analyzer according to theexpression of M_(i) in advance; Step
 3. Set up the non-adaptivemeasurement apparatus, use the setting parameters of the quantum devicedetermined in step 1 to generate copies of the specific state σ_(i) oneby one, and perform the non-adaptive projective measurements {p₁P₁,p₂P₂, . . . , p_(i)P_(i), . . . , p_(N)P_(N)} on σ_(i). At the sametime, execute coincidence counts through the time correlation countingmodule, and record the timetag data of each projective measurement baseP_(i); Build the adaptive measurement apparatus using anexternally-triggered instrument, characterize and set the triggerinstrument according to the parameters of the adaptive measurementcalculated in step 2, and use the electrical signal output from thelogic array to control the triggering device to implement classicalcommunication between the two subsystems, and perform the adaptivemeasurement sets {p₁T₁, p₂T₂, . . . , p_(i)T_(i), . . . , p_(N)T_(N)};According to the expression of adaptive measurement, the triggeringdevice can be adjusted independently to realize the respectiveprojective measurement. The overall adaptive measurement T_(i) can beperformed by combing the different triggering devices. Realize real-timecontrol of projective base of particle B according to the measurementresult of particle A, and record the timetag data under each projectivebase T_(i); According to the timetag data, make programming to extract asingle coincidence count. The time stamp corresponding to each channelis separated firstly, and then the time is sliced. The coincidence countis scanned from the initial time slice to the final time slice; If thereis only one coincidence count, record the corresponding coincidencechannel, and iterate to the next time slice until all single coincidencecounts are found, and record all the time slices and coincidence channeldata corresponding to each single coincidence count, and save them inthe form of a data table by column. At the same time, if the coincidencechannel falls on the channel corresponding to the successful projectivemeasurement, it is recorded as success 1, otherwise it is recorded asfailure 0, and the data of success 1 and failure 0 are also stored as acolumn in the data table; Step 4: The projective measurement P_(i)/T_(i)is selected randomly according to the probability p_(i) corresponding toeach projective base in the measurement sets, which is used to simulatethe random measurement process, and then the statistical process of taskA and task B is performed; Task A performs tests on the generatedquantum state from front to back, and obtains the projective measurementresults of each copy according to the success probability of eachcoincidence channel. If success, the data 1 is recorded, while 0 isrecorded for failure. When 0 appears for the first time, the subsequentprojective measurement is terminated, and the index N_(first) of thefirst failure event is recorded. This process is cycled for 10000rounds. In each round, the index of the occurrence of first failureevent is recorded. Finally, a geometric probability distribution aredetermined for the index that fails for the first time. The dataextraction for the first failure event can be made simultaneously forboth non-adaptive and adaptive measurements; Task B fixes the number oftests N, and selects P_(i)/T_(i) from the measurement sets withprobability p_(i) each time. Likewise, the measurement result isobtained as 1 for success or 0 for failure through the coincidencecount. Finally, the binary sequence 11101011011 . . . 1 is obtained.After making statistics on the sequence, the number of success eventsm_(pass) in the N times can be obtained. This process for extracting thenumber of success events in N times can also be performed simultaneouslyfor the non-adaptive and adaptive measurements; Step
 5. The firstfailure index N_(first) in task A will constitute a geometricdistribution, and the cumulative probability is the confidence:$\delta_{A} = {\sum\limits_{N_{first} = 1}^{n_{\exp}}{\Pr\left( N_{first} \right)}}$Calculate the number of measurements n_(exp) required for the cumulativeprobability to reach 90%. Fitting the probability distribution to obtainan estimate of the infidelity of the quantum state ∈_(exp). According tothe fitted ∈_(exp), a suitable ϵ parameter is given, and the theoreticalsuccess probability μ=1−Δ_(∈) 0 is obtained. The equipment is dividedinto two categories according to the chosen ϵ, one is Case 1:

ψ|σ_(i)|ψ

>1−∈, the other is Case 2:

ψ|σ_(i)|ψ

≤1−∈, which is in correspondence with the results m_(pass)>μN andm_(pass)<μN, respectively. Then the Chernoff bound in probabilitytheory: $\delta \equiv e^{- {{ND}({\frac{m_{pass}}{N}{}\mu})}}$ isused to estimate the variation of confidence level 1−δ and fidelity 1−ϵversus the number of copies of quantum states N.